Abstract

General solutions for a four-flux radiative transfer model, derived
from the radiative transfer equation and based on Lorenz–Mie
scattering and absorption parameters, have been obtained. Forward
and backward average path-length parameters have been considered as
well as forward-scattering ratios for diffuse anisotropic radiation
going into the forward and the backward hemispheres. The reported
solutions are generalizations of those obtained by Maheu et
al. [Appl. Opt.23, 3353–3362
(1984)]. Compared with the generalized solutions, numerical
calculations indicate that the δ-Eddington approximation and the
standard four-flux model of Maheu et al. overestimate the
collimated–diffuse reflectance of particulate coatings, whereas these
models give similar results in the case of collimated–diffuse
transmittance.

Optical thickness dependence of the collimated–diffuse
reflectance of an unsupported thick polyethylene film (h =
50 μm) containing titanium dioxide pigments at a
particle volume fraction f = 0.05. The free-space
wavelength was set to 0.55 μm. The refractive index of the
pigments is 2.50 + i0.0, and the refractive index of
the matrix was set to 1.50. MLG, values obtained from the four-flux
model of Maheu, Letoulouzan, and Gouesbet,10 where we
evaluate ξ and σd by using the extended
Hartel theory of Vargas and Niklasson13; GFF, values
calculated from the generalized four-flux model described throughout
the paper; EDD, reflectance values obtained from the integrated
δ-Eddington approximation. Boundary reflections have been
neglected.